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Problem 389

Algebra Difficulty 2.5 Multiple choice CEMC Pascal · Canada · 2015

At the end of the year 2000, Steve had $100 and Wayne had$ 10 000. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?

Pick one

Official solutions — 2

Solution 1

Year
Steve
Wayne

$2000$
$\$100$
$\$10\,000$

$2001$
$\$200$
$\$5000$

$2002$
$\$400$
$\$2500$

$2003$
$\$800$
$\$1250$

$2004$
$\$1600$
$\$625$

Therefore, 2004 is the first time at which Steve has more money than Wayne at the end of the year.

Solution 2

We make a table of the total amount of money that each of Steve and Wayne have at the end of each year. After the year 2000, each entry in Steve’s column is found by doubling the previous entry and each entry in Wayne’s column is found by dividing the previous entry by 2. We stop when the entry in Steve’s column is larger than that in Wayne’s column: $\text{Year Steve Wayne 2000 100 10 ,000 2001 200 5000 2002 400 2500 2003 800 1250 2004 1600 625}$ Therefore, the end of 2004 is the first time at which Steve has more money than Wayne at the end of the year.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.